Random Integer

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snow5379
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Random Integer

Postby snow5379 » Tue Apr 12, 2011 7:20 pm UTC

Imagine a hypothetical universe where there are an infinite number of planets. I can imagine a god of some sort choosing one to create life on. I can also imagine a different god labeling each planet with a number. At some point these two gods meet up and start talking:

God A: "I created life on this planet. I still haven't thought of a name for it though."
God B: "I just use numbers. I call that planet number..."

With our current understanding of mathematics whatever number the second god mentions the first would have expected a higher one. However I don't think this is the case and feel that the hypothetical gods would have a better understanding of mathematics than we do. Simply put I've started to question the nature of infinite numbers and the paradoxes that arise from them.

Again let's look back at these hypothetical gods. They both have defined a meter. The first god's meter is infinitely larger than the second god's meter and the second god's meter is infinitely smaller than the firsts. Both see their own meter as finite and the other as infinitely small or large. I look at such things and I'm just not seeing much difference at all between finite and infinite numbers... to me they just look like different points of view.

As I understand it choosing a random integer isn't much different between choosing a number between 0 and 1 if you can view infinitesimal, finite, and infinite numbers as being... well... just different points of view. Also I realize this only works when you tack on units (because clearly one dog is less than an infinite number of dogs).

I think the second could will have labeled the planet with life on it with some infinite number and somewhere in the universe there is a first and last planet... even if there are an infinite number of them. It's similar to, say, how there are an infinite number of reals between 0 and 1 and yet there's clearly a first and last.

It may be that way we define integers is what causes all the paradoxes we see in mathematics and we keep blaming it on everything else and saying things like "you can't divide by this or do that"... which is perfectly valid because math is simply a definition game... but when you apply this definition game to the real world or even a hypothetical world maybe you should consider redefining things when obvious problems arise?

Yesila
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Re: Random Integer

Postby Yesila » Wed Apr 13, 2011 6:01 am UTC

snow5379 wrote:...
I think the second could will have labeled the planet with life on it with some infinite number and somewhere in the universe there is a first and last planet... even if there are an infinite number of them. It's similar to, say, how there are an infinite number of reals between 0 and 1 and yet there's clearly a first and last....


There is not clearly a first and last. There is clearly a smallest and Largest, but that is not the same things as first and last. Using the words First and Last also suggests the existence of such things as a Second and Third and Fifty-seventh etc... which in turn suggests that the reals are countable... which they aren't

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antonfire
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Re: Random Integer

Postby antonfire » Wed Apr 13, 2011 6:13 am UTC

snow5379 wrote:It may be that way we define integers is what causes all the paradoxes we see in mathematics and we keep blaming it on everything else and saying things like "you can't divide by this or do that"... which is perfectly valid because math is simply a definition game... but when you apply this definition game to the real world or even a hypothetical world maybe you should consider redefining things when obvious problems arise?
Yes, you should. If you can find an equally simple, descriptive, useful system where such problems don't arise, publish.
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Re: Random Integer

Postby DavCrav » Wed Apr 13, 2011 9:17 am UTC

Yesila wrote:
snow5379 wrote:...
I think the second could will have labeled the planet with life on it with some infinite number and somewhere in the universe there is a first and last planet... even if there are an infinite number of them. It's similar to, say, how there are an infinite number of reals between 0 and 1 and yet there's clearly a first and last....


There is not clearly a first and last. There is clearly a smallest and Largest, but that is not the same things as first and last. Using the words First and Last also suggests the existence of such things as a Second and Third and Fifty-seventh etc... which in turn suggests that the reals are countable... which they aren't


If you include 0 and 1 there are...

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Kurushimi
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Re: Random Integer

Postby Kurushimi » Wed Apr 13, 2011 1:32 pm UTC

Yesila said "there is not always a first and a last". You gave one example of there being a first and a last, (i.e. numbers between 0 and 1 inclusive) (at least, that's what I think you were referring to). Now, let's say I take the set (0,1), numbers between 0 and 1 exclusive. Can you now point out the first and last numbers in this set?

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silverhammermba
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Re: Random Integer

Postby silverhammermba » Wed Apr 13, 2011 2:07 pm UTC

snow5379 wrote:God A: "I created life on this planet. I still haven't thought of a name for it though."
God B: "I just use numbers. I call that planet number..."

With our current understanding of mathematics whatever number the second god mentions the first would have expected a higher one. However I don't think this is the case and feel that the hypothetical gods would have a better understanding of mathematics than we do. Simply put I've started to question the nature of infinite numbers and the paradoxes that arise from them.

Not quite. If B is numbering planets sequentially and A has no information about how far along in the sequence B is, then A has literally no chance of guessing what number B will choose. Since B could be anywhere in his numbering sequence, A would need to give the natural numbers a uniform distribution and then select one. But that is mathematically impossible! Even if B tells A "I've already numbered planets 1 through 100 (but possibly more)", A still has absolutely no chance. He would need to put a uniform distribution on {101,102,...} but that's clearly bijective with the natural numbers so we're back to square one!

Cranica
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Re: Random Integer

Postby Cranica » Wed Apr 13, 2011 3:15 pm UTC

That example would work on [0,1], too. Removing any finite set from an infinite one will never get you down to a finite set, but clearly probability on [0,1] (or (0,1)) is well-defined. The issue as I understand it isn't the size of the set, it's that the integers are discrete.

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Re: Random Integer

Postby Mindworm » Wed Apr 13, 2011 5:09 pm UTC

The issue as I understand it isn't the size of the set, it's that the integers are discrete.

No, it's not. There are perfectly fine probability distributions on the integers, just no uniform one. And neither is there one on [0, 1]. Once your set is infinite, there is no uniform distribution.
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Re: Random Integer

Postby ++$_ » Wed Apr 13, 2011 5:37 pm UTC

Mindworm wrote:
The issue as I understand it isn't the size of the set, it's that the integers are discrete.

No, it's not. There are perfectly fine probability distributions on the integers, just no uniform one. And neither is there one on [0, 1]. Once your set is infinite, there is no uniform distribution.
Are you trying to say there is no uniform distribution on the interval [0,1]?

Because there definitely is one.

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Re: Random Integer

Postby Aiwendil » Wed Apr 13, 2011 5:45 pm UTC

And neither is there one on [0, 1].


Huh? That would mean there's no such thing as a continuous uniform distribution, which I'm pretty sure is not true.

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thedufer
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Re: Random Integer

Postby thedufer » Wed Apr 13, 2011 8:21 pm UTC

Aiwendil wrote:
And neither is there one on [0, 1].


Huh? That would mean there's no such thing as a continuous uniform distribution, which I'm pretty sure is not true.


No, he's definitely right on this one. I'm fairly certain that if there was a continuous uniform distribution, you'd be able to map that to a uniform distribution over the integers (since the cardinality of [0, 1] is larger than the cardinality of the integers).

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Re: Random Integer

Postby letterX » Wed Apr 13, 2011 8:28 pm UTC

thedufer wrote:
Aiwendil wrote:
And neither is there one on [0, 1].


Huh? That would mean there's no such thing as a continuous uniform distribution, which I'm pretty sure is not true.


No, he's definitely right on this one. I'm fairly certain that if there was a continuous uniform distribution, you'd be able to map that to a uniform distribution over the integers (since the cardinality of [0, 1] is larger than the cardinality of the integers).


No, he is definitely not right. There is absolutely a uniform distribution on the unit interval. Cardinality has, unfortunately, nothing to do with the matter. This is really more a matter of topology, and the topologies of the unit interval and the integers are vastly different.

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Dason
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Re: Random Integer

Postby Dason » Thu Apr 14, 2011 1:44 am UTC

I am kind of wondering how somebody could think there couldn't be a uniform distribution on [0,1]?
double epsilon = -.0000001;

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MartianInvader
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Re: Random Integer

Postby MartianInvader » Thu Apr 14, 2011 1:35 pm UTC

It actually is sort of a cardinality thing. You can't ever have a uniform, countably additive distribution on a countable set - the probability of getting each element would have to be a number that, when added to itself countably many times, gives you 1. Of course, no such number exists.

It makes more sense to drop the "countably additive" requirement from the definitions of measure and distribution when talking about the integers, and instead require that they only be finitely additive.

Then you can have a uniform probability measure on the integers, and statements like "The probability that a random integer is even is 1/2" are well-defined. (And typically true!)
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mike-l
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Re: Random Integer

Postby mike-l » Thu Apr 14, 2011 8:39 pm UTC

MartianInvader wrote:Then you can have a uniform probability measure on the integers, and statements like "The probability that a random integer is even is 1/2" are well-defined. (And typically true!)


What's your definition of uniform? The best I can come up with that gives anything of interest is 'translation invariant'.
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jestingrabbit
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Re: Random Integer

Postby jestingrabbit » Thu Apr 14, 2011 9:33 pm UTC

mike-l wrote:What's your definition of uniform? The best I can come up with that gives anything of interest is 'translation invariant'.


This is the right definition of uniform. Attaching a few more sensible properties gets you Haar measures and such measures are (essentially) unique.
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MartianInvader
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Re: Random Integer

Postby MartianInvader » Thu Apr 14, 2011 9:43 pm UTC

Well, except that Haar measures are again typically defined on uncountable spaces, as they are countably additive. I'm thinking more along the lines of finitely-additive "measures" defined on amenable discrete groups, like the integers.

Of course, these "measures" (in quotes because they're not countably additive) are far from unique; I don't know the exact cardinality but I'm pretty sure there's at least [imath]2^{\omega}[/imath] of them on the integers.
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Cranica
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Re: Random Integer

Postby Cranica » Fri Apr 15, 2011 6:07 am UTC

MartianInvader wrote:It actually is sort of a cardinality thing. You can't ever have a uniform, countably additive distribution on a countable set - the probability of getting each element would have to be a number that, when added to itself countably many times, gives you 1. Of course, no such number exists.

It makes more sense to drop the "countably additive" requirement from the definitions of measure and distribution when talking about the integers, and instead require that they only be finitely additive.

Then you can have a uniform probability measure on the integers, and statements like "The probability that a random integer is even is 1/2" are well-defined. (And typically true!)


How does one do this formally? Is it limit based (i.e., something like: p = lim(n->infinity) ([naturals <= n satisfying property]/n))? That would seem to imply statements like "the probability of a random natural being prime is 0", since the prime counting function approaches ln(n)/n in the limit, which seems odd since the probability is understood and finite for any actual interval.

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jestingrabbit
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Re: Random Integer

Postby jestingrabbit » Fri Apr 15, 2011 6:57 am UTC

Cranica wrote:How does one do this formally? Is it limit based (i.e., something like: p = lim(n->infinity) ([naturals <= n satisfying property]/n))? That would seem to imply statements like "the probability of a random natural being prime is 0", since the prime counting function approaches ln(n)/n in the limit, which seems odd since the probability is understood and finite for any actual interval.


That's pretty much exactly how you do it. The groups that you can do this with are called Amenable groups and the generalisation of the sequence of sets [imath]\{\{0,\ldots ,n\}\}_{n\in N}[/imath] are called Følner sequences. Of course the natural numbers aren't a group, but you can apply the same ideas pretty easily.

I think that your argument about the primes isn't that great. I think that the "size" of a set should have a few properties. Firstly, any finite amount of tinkering to the set shouldn't change its size and secondly if [imath]A\subseteq B[/imath] then the size of A should be less than or equal to the size of B. Specific to the natural numbers, I would require that if a and b are natural, then a set of the form [imath]\{ an + b | n\in N\}[/imath] should have size 1/a. With those three things, and additivity ie [imath]m(A\cup B) \leq m(A) + m(B)[/imath], you can pretty quickly arrive at the conclusion that the size of the primes is 0.

Moving from the size of a set to calling it a probability isn't entirely justified but moving from the prime number theorem to a statement about probabilities is less justified imo.
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